Date: Apr 25, 2013 5:31 AM
Author: quasi
Subject: Re: Onto [0,1]
quasi wrote:

>William Elliot wrote:

>>baclesback wrote:

>>>

>>> More specifically, use the representation of x in C Cantor

>>> set in base 3 with only 0's and 2's in the expansion of 3,

>>> and map

>>>

>>> f: x=0.a1a2..... ---> 0.b1b2.......

>>>

>>> Wheref(bi)= 0 , if ai=0 , f(bi)=1 , if ai=2 .

>>>

>>> This f is continuous, but not absolutely continuous ( which

>>> preserves sets of measure zero. )

>>

>>Why is f continuous?

>

>Because f is increasing.

I meant:

Because f is _strictly_ increasing.

>If X,Y are subsets of R, and f: X -> Y is a monotonic function,

I meant:

If X,Y are subsets of R, and f: X -> Y is a _strictly_ monotonic

function,

>then f is continuous (with respect to the relative topologies on

>X and Y inherited from R).

quasi